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A real solution of the functional equationϕ(x + ψ(y − x)) = f(x) + g(y) + h(x)k(y) on a setΩ ⊂ ℝ2 is a 6-tuple (f, g, h, k, ϕ, ψ) of real valued functions such that the equation is identically fulfilled onΩ. Except for cases known before—e.g. whenψ is linear—we present all real solutions in an...
Gauss proved Seeber's Theorem, that the determinant of a reduced positive definite ternary quadratic form is at least half the product of its diagonal coefficients, by means of two determinantal identities whose origin has remained unclear. We examine Gauss's method from a general standpoint, as...
There exists a Borel set C of product Lebesgue measure one in the Hilbert cube having the property that, for every measure preserving transformationT of the unit interval, allT-orbits contained inC originate from a zero set. This settles an infinite dimensional version of a problem raised by Th....
In this paper, we study the convergence of formal power series solutionsψ of functional equations of the form∑P
(x) denotes thek-th iterate of the functionϕ.
The following theorem holds true.
Using a generalized Cauchy functional equation we show that some well-known characterizations of inner product spaces, such as those of Jordan—von Neumann, Johnson, and Rassias, can be proved without use of the triangle inequality.
In the paper a discrete analog to the Volterra nonlinear integral equation is discussed. Weighted norms are used to find sufficient conditions that all solutions of such equations are elements of anl
Letn ⩾ 2 be an integer and letω be a primitiven-th root of unity. Let us call a functionh: ℂ → ℂ to be of typej(0 ⩽ j ⩽ n − 1), ifh(ωx) = ω
h(x) holds for allx ∈ ℂ.
Letf be a map from a groupG into an abelian groupH satisfyingf(xy) + f(xy
−1) = 2f(x), f(e) = 0, wherex, y ∈ G ande is the identity inG. A set of necessary and sufficient conditions forS(G, H) = Hom(G, H) is given whenG is abelian, whereS(G, H) denotes all the solutions of the functional...
In the present note we prove that every functionf: (0, ∞) → [0, ∞) satisfying the inequalityaf(s) + bf(t) ⩽ f(as + bt), s, t > 0, for somea andb such that 0 0). This improves our recent result in , where the inequality is assumed to hold for...
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